Geoscience Reference
In-Depth Information
This equation is hold when the moments
m
n
are finite, i.e.
|
m
n
|
<
∞
.
The moment method claims that using the moment of the random variable
X
, the PDF of
X
is
completely determined. So if we have:
E
X
n
E
X
k
,
=
∀
lim
n
k
(4)
→
∞
then, the sequence
has the same distribution as the
X
. we use (4) for parameter
estimation, i.e. The left side of the equation is obtained statically, and the right side is
calculated analytically. These two sides should be equal.
{
X
n
}
To begin our discussion, a model should be considered for our signals. Next section is focused
on finding the suitable model.
3. Signal model
We consider the baseband representation of the received signal, which can be expressed as
the sum of the desired signal component and non-stationary background noise. The signal
component is represented by the linear sum of many non-coherent waveforms whose arrivals
at the receiver are governed by a Poisson process [Zabin & Wright, 1994]. The receiver includes
two sensors to measure the received signal in presence of background noise:
y
1
(
t
) =
s
(
t
) +
ω
1
(
t
)
,
)
=
− τ
)
ε
)
+
ω
2
(
y
2
(
t
s
(
t
exp
(
j
2
π
t
t
)
,
(5)
where
τ
and
ε
denote the time delay and Doppler respectively, and
s
is the desired received
signal modeled at any time instance
t
to follow a real N-mode Gaussian mixture distribution
[Isaksson et al., 2001]:
(
t
)
i
=
1
p
i
N
μ
s
i
,
σ
s
i
.
N
s
(
t
)
∼
(6)
The processes
ω
1
(
t
)
and
ω
2
(
t
)
are real zero-mean additive white Gaussian noises (AWGN)
2
ω
1
2
with powers of
ω
2
respectively. These powers are not constant in practice due to
nonhomogeneous environment, but are assumed as random variates which are estimated
subsequently.
σ
and
σ
The signal and noise are supposed to be uncorrelated, but the noises
ω
1
(
t
)
and
ω
2
(
t
)
are possibly correlated.
4. Parameter estimation
In this section, for a random variable
X
, the moment generating function (MGF),
M
x
(
u
)
, and
its asymptotic series are used to determine the moments
m
xi
:
u
2
m
x
2
2!
e
uX
M
x
(
u
)=
E
(
)=
1
+
um
x
1
+
+
···
,
u
→
0.
(7)
4.1 Time delay estimation
The statistical properties of the signal and noise which are represented in (5) are known.
Therefore, their MGF is available, by assuming finite moments of signal and noise. Although
the signal follows a Gaussian mixture distribution, the conglomerate effect of the time delay
